TensorSum

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 Other toolboxes required TensorSum Computes a vector or operator from its tensor decomposition none FilterNormalFormOperatorSchmidtDecompositionSchmidtDecomposition Basic operation

TensorSum is a function that computes a vector or operator from its tensor decomposition. It acts as an inverse of the OperatorSchmidtDecomposition and SchmidtDecomposition functions, but also works for multipartite decompositions.

Syntax

• TS = TensorSum(A1,A2,...)
• TS = TensorSum(S,A1,A2,...)

Argument descriptions

• A1,A2,...: Either matrices or cells containing matrices (and they should all be the same: either all matrices or all cells). If they are matrices, then the k-th column of each Ai will be tensored together for all k, and then the sum over k will be taken at the end. If they are cells, then the k-th element of each Ai will be tensored together for all k, and then the sum over k will be taken at the end.
• S (optional, default [1,1,...,1]): A vector of weights (such as Schmidt coefficients) that will be applied when summing the terms at the end of the computation.

Examples

Inverse of SchmidtDecomposition

This function acts as an inverse of the SchmidtDecomposition function:

```>> vec = [1,2,3,4];
>> [s,u,v] = SchmidtDecomposition(vec);
>> TensorSum(s,u,v)

ans =

1.0000
2.0000
3.0000
4.0000```

Inverse of OperatorSchmidtDecomposition

This function also acts as an inverse of the OperatorSchmidtDecomposition function:

```>> X = reshape(1:16,4,4)'

X =

1     2     3     4
5     6     7     8
9    10    11    12
13    14    15    16

>> [s,U,V] = OperatorSchmidtDecomposition(X);
>> TensorSum(s,U,V)

ans =

1.0000    2.0000    3.0000    4.0000
5.0000    6.0000    7.0000    8.0000
9.0000   10.0000   11.0000   12.0000
13.0000   14.0000   15.0000   16.0000```

Source code

Click on "expand" to the right to view the MATLAB source code for this function.

1. `%%  TENSORSUM  Computes a vector or operator from its tensor decomposition`
2. `%   This function can be called in two ways:`
3. `%   TS = TensorSum(A1,A2,...)`
4. `%   TS = TensorSum(S,A1,A2,...)`
5. `%`
6. `%   Each term Ai is either a matrix or a cell containing matrices (and they`
7. `%   should all be the same: either all matrices or all cells). If they are`
8. `%   matrices, then the k-th column of each Ai will be tensored together for`
9. `%   all k, and then the sum over k will be taken at the end. If they are`
10. `%   cells, then the k-th element of each Ai will be tensored together for`
11. `%   all k, and then the sum over k will be taken at the end.`
12. `%`
13. `%   If S is provided, it is a vector of weights (such as Schmidt`
14. `%   coefficients) that will be applied when summing the terms at the end of`
15. `%   the computation.`
16. `%`
17. `%   This function acts as an inverse of the SchmidtDecomposition and`
18. `%   OperatorSchmidtDecomposition functions. For example, if [S,U,V] =`
19. `%   SchmidtDecomposition(VEC) then VEC = TensorSum(S,U,V).`
20. `%`
21. `%   URL: http://www.qetlab.com/TensorSum`
22. ` `
23. `%   requires: nothing`
24. `%   author: Nathaniel Johnston (nathaniel@njohnston.ca)`
25. `%   package: QETLAB`
26. `%   last updated: December 19, 2014`
27. ` `
28. `function ts = TensorSum(a1,a2,varargin)`
29. ` `
30. `% Determine the number of terms in the tensor decomposition.`
31. `if(iscell(a2))`
32. `    len = length(a2);`
33. `else`
34. `    len = size(a2,2);`
35. `end`
36. ` `
37. `% Determine whether or not a vector of coefficients was provided, then read`
38. `% in the input arguments.`
39. `if(iscell(a1) || min(size(a1)) > 1 || len == 1)`
40. `    s = ones(len,1);`
41. `    A = [{a1} {a2} varargin];`
42. `else`
43. `    s = a1;`
44. `    A = [{a2} varargin];`
45. `end`
46. ` `
47. `% Get some more preliminary values (dimensions, etc.) and then convert`
48. `% everything into cell matrices (instead of matrices of column vectors) if`
49. `% necessary.`
50. `num_parties = length(A);`
51. `dim = zeros(2,num_parties);`
52. `for j = 1:num_parties`
53. `    if(~iscell(A{j}))`
54. `        A{j} = mat2cell(A{j},size(A{j},1),ones(1,size(A{j},2)));`
55. `    end`
56. `    dim(:,j) = size(A{j}{1}).';`
57. `end`
58. ` `
59. `% Finally, actually compute the tensor sum.`
60. `ts = sparse(prod(dim(1,:)),prod(dim(2,:)));`
61. `for j = 1:len`
62. `    ts_tmp = A{1}{j};`
63. `    for k = 2:num_parties`
64. `        ts_tmp = kron(ts_tmp,A{k}{j});`
65. `    end`
66. `    ts = ts + s(j)*ts_tmp;`
67. `end`